# Monoidal structure on left dg-modules over a brace algebra

Relating to my other question: Modules over Hopf Algebras and $$E_2$$-algebras

Preliminary: Let $$A$$ be an associative dg-algebra that is also an algebra over the brace operad. Let $$M$$ and $$N$$ be left dg-modules and assume we have previously constructed a quasi iso.

$$f_k:A^{\otimes k}\to A^{op}$$ from the dg-algebra structure of $$A$$ to its opposite algebra based on the brace structure (which makes $$A$$ homotopy commutative).

We can construct the chain complex (ignore signs for now)

$$M \otimes N:= M\otimes_k \mathfrak B (A) \otimes_k N$$ $$\partial (m\otimes a_1 \otimes \cdots \otimes a_n \otimes n):= dm\otimes \cdots \otimes a_n \otimes n +m\otimes d(a_1)\otimes \cdots \otimes n + \cdots +$$ $$m \otimes \cdots \otimes a_n \otimes dn +$$ $$a_1\cdot m\otimes a_2 \otimes \cdots \otimes n + f_2(a_1,a_2)\cdot m \otimes \cdots \otimes n+ \cdots +f_n(a_1 ,\ldots ,a_n)\cdot m\otimes n +$$ $$m\otimes a_1 a_2 \otimes \cdots \otimes n + \cdots + m \otimes a_1 \otimes \cdots \otimes a_{n-1} \otimes a_n \cdot n$$

where $$\mathfrak B (A)$$ is the bar complex of $$A$$. This is my attempt to turn the category of left modules over $$A$$ into a monoidal (or maybe monoidal up to homotopy) category.

The problem then becomes that this new chain complex is not a left module, at least with the logical choice of action on the m component of the tensor. The action of $$A$$ on the first component does not respect the differential of $$M\otimes N$$ and so we don't get a left dg-module out.

Question1: Is there a way to change this construction to ensure that what we get out is a true left dg-module over $$A$$?

Question2: If $$M$$ and $$N$$ were instead left $$A_{\infty}$$ modules over $$A$$, can we (easily?) alter the construction to make the result an $$A_{\infty}$$ module?

Question3: If we collapse everything by taking cohomology, then will the result be a left module over $$H^*(A)$$ (which is now commutative). Do we just get Tor back? Is there any way to get anything more than just classical results about commutative algebra by doing this?

Thanks, -Matt